What kind of output should be used for predicting angles in DNNs?
As you said, the first option is not very suitable due to the cyclic nature of the angles. However, if you don't mind discretizing the values, you could represent the output as a binary vector.

A variant of the second option seems perfect to me. You may output a 2D vector and use that vector's angle as output. You'll probably need a regularizer for the vector's norm, but that's it.